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  <div class="section" id="numpy-linalg-svd">
<h1>numpy.linalg.svd<a class="headerlink" href="#numpy-linalg-svd" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.linalg.svd">
<code class="sig-prename descclassname">numpy.linalg.</code><code class="sig-name descname">svd</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">full_matrices=True</em>, <em class="sig-param">compute_uv=True</em>, <em class="sig-param">hermitian=False</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/linalg/linalg.py#L1458-L1640"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.linalg.svd" title="Permalink to this definition">¶</a></dt>
<dd><p>Singular Value Decomposition.</p>
<p>When <em class="xref py py-obj">a</em> is a 2D array, it is factorized as <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">&#64;</span> <span class="pre">np.diag(s)</span> <span class="pre">&#64;</span> <span class="pre">vh</span>
<span class="pre">=</span> <span class="pre">(u</span> <span class="pre">*</span> <span class="pre">s)</span> <span class="pre">&#64;</span> <span class="pre">vh</span></code>, where <em class="xref py py-obj">u</em> and <em class="xref py py-obj">vh</em> are 2D unitary arrays and <em class="xref py py-obj">s</em> is a 1D
array of <em class="xref py py-obj">a</em>’s singular values. When <em class="xref py py-obj">a</em> is higher-dimensional, SVD is
applied in stacked mode as explained below.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>a</strong><span class="classifier">(…, M, N) array_like</span></dt><dd><p>A real or complex array with <code class="docutils literal notranslate"><span class="pre">a.ndim</span> <span class="pre">&gt;=</span> <span class="pre">2</span></code>.</p>
</dd>
<dt><strong>full_matrices</strong><span class="classifier">bool, optional</span></dt><dd><p>If True (default), <em class="xref py py-obj">u</em> and <em class="xref py py-obj">vh</em> have the shapes <code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">M,</span> <span class="pre">M)</span></code> and
<code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">N,</span> <span class="pre">N)</span></code>, respectively.  Otherwise, the shapes are
<code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">M,</span> <span class="pre">K)</span></code> and <code class="docutils literal notranslate"><span class="pre">(...,</span> <span class="pre">K,</span> <span class="pre">N)</span></code>, respectively, where
<code class="docutils literal notranslate"><span class="pre">K</span> <span class="pre">=</span> <span class="pre">min(M,</span> <span class="pre">N)</span></code>.</p>
</dd>
<dt><strong>compute_uv</strong><span class="classifier">bool, optional</span></dt><dd><p>Whether or not to compute <em class="xref py py-obj">u</em> and <em class="xref py py-obj">vh</em> in addition to <em class="xref py py-obj">s</em>.  True
by default.</p>
</dd>
<dt><strong>hermitian</strong><span class="classifier">bool, optional</span></dt><dd><p>If True, <em class="xref py py-obj">a</em> is assumed to be Hermitian (symmetric if real-valued),
enabling a more efficient method for finding singular values.
Defaults to False.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.17.0.</span></p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>u</strong><span class="classifier">{ (…, M, M), (…, M, K) } array</span></dt><dd><p>Unitary array(s). The first <code class="docutils literal notranslate"><span class="pre">a.ndim</span> <span class="pre">-</span> <span class="pre">2</span></code> dimensions have the same
size as those of the input <em class="xref py py-obj">a</em>. The size of the last two dimensions
depends on the value of <em class="xref py py-obj">full_matrices</em>. Only returned when
<em class="xref py py-obj">compute_uv</em> is True.</p>
</dd>
<dt><strong>s</strong><span class="classifier">(…, K) array</span></dt><dd><p>Vector(s) with the singular values, within each vector sorted in
descending order. The first <code class="docutils literal notranslate"><span class="pre">a.ndim</span> <span class="pre">-</span> <span class="pre">2</span></code> dimensions have the same
size as those of the input <em class="xref py py-obj">a</em>.</p>
</dd>
<dt><strong>vh</strong><span class="classifier">{ (…, N, N), (…, K, N) } array</span></dt><dd><p>Unitary array(s). The first <code class="docutils literal notranslate"><span class="pre">a.ndim</span> <span class="pre">-</span> <span class="pre">2</span></code> dimensions have the same
size as those of the input <em class="xref py py-obj">a</em>. The size of the last two dimensions
depends on the value of <em class="xref py py-obj">full_matrices</em>. Only returned when
<em class="xref py py-obj">compute_uv</em> is True.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>LinAlgError</strong></dt><dd><p>If SVD computation does not converge.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 1.8.0: </span>Broadcasting rules apply, see the <a class="reference internal" href="../routines.linalg.html#module-numpy.linalg" title="numpy.linalg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.linalg</span></code></a> documentation for
details.</p>
</div>
<p>The decomposition is performed using LAPACK routine <code class="docutils literal notranslate"><span class="pre">_gesdd</span></code>.</p>
<p>SVD is usually described for the factorization of a 2D matrix <img class="math" src="../../_images/math/211284f68205c3e66773eaf026f32a0acdd3dfb3.svg" alt="A"/>.
The higher-dimensional case will be discussed below. In the 2D case, SVD is
written as <img class="math" src="../../_images/math/20186250ac016f1d53ee5e5269bd0dca2357e051.svg" alt="A = U S V^H"/>, where <img class="math" src="../../_images/math/6f58e19109d7814530308e7988bc79eb52360ed3.svg" alt="A = a"/>, <img class="math" src="../../_images/math/0261c269f6ccae7b2f8251a8bffa4055cea0021c.svg" alt="U= u"/>,
<img class="math" src="../../_images/math/e1302c3cd9c211d641aeec1ec0cba8bdafbfa708.svg" alt="S= \mathtt{np.diag}(s)"/> and <img class="math" src="../../_images/math/da38143ce067d4d5c490563b7b05874a098daad9.svg" alt="V^H = vh"/>. The 1D array <em class="xref py py-obj">s</em>
contains the singular values of <em class="xref py py-obj">a</em> and <em class="xref py py-obj">u</em> and <em class="xref py py-obj">vh</em> are unitary. The rows
of <em class="xref py py-obj">vh</em> are the eigenvectors of <img class="math" src="../../_images/math/eb92fc1ff2c7dc528cb2e40ae997408b2c4cff94.svg" alt="A^H A"/> and the columns of <em class="xref py py-obj">u</em> are
the eigenvectors of <img class="math" src="../../_images/math/576a4180b76c948f7e9dc20f22880e39fba29668.svg" alt="A A^H"/>. In both cases the corresponding
(possibly non-zero) eigenvalues are given by <code class="docutils literal notranslate"><span class="pre">s**2</span></code>.</p>
<p>If <em class="xref py py-obj">a</em> has more than two dimensions, then broadcasting rules apply, as
explained in <a class="reference internal" href="../routines.linalg.html#routines-linalg-broadcasting"><span class="std std-ref">Linear algebra on several matrices at once</span></a>. This means that SVD is
working in “stacked” mode: it iterates over all indices of the first
<code class="docutils literal notranslate"><span class="pre">a.ndim</span> <span class="pre">-</span> <span class="pre">2</span></code> dimensions and for each combination SVD is applied to the
last two indices. The matrix <em class="xref py py-obj">a</em> can be reconstructed from the
decomposition with either <code class="docutils literal notranslate"><span class="pre">(u</span> <span class="pre">*</span> <span class="pre">s[...,</span> <span class="pre">None,</span> <span class="pre">:])</span> <span class="pre">&#64;</span> <span class="pre">vh</span></code> or
<code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">&#64;</span> <span class="pre">(s[...,</span> <span class="pre">None]</span> <span class="pre">*</span> <span class="pre">vh)</span></code>. (The <code class="docutils literal notranslate"><span class="pre">&#64;</span></code> operator can be replaced by the
function <code class="docutils literal notranslate"><span class="pre">np.matmul</span></code> for python versions below 3.5.)</p>
<p>If <em class="xref py py-obj">a</em> is a <code class="docutils literal notranslate"><span class="pre">matrix</span></code> object (as opposed to an <code class="docutils literal notranslate"><span class="pre">ndarray</span></code>), then so are
all the return values.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="n">j</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="n">j</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
</pre></div>
</div>
<p>Reconstruction based on full SVD, 2D case:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vh</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">full_matrices</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">vh</span><span class="o">.</span><span class="n">shape</span>
<span class="go">((9, 9), (6,), (6, 6))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">6</span><span class="p">]</span> <span class="o">*</span> <span class="n">s</span><span class="p">,</span> <span class="n">vh</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">smat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">9</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">complex</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">smat</span><span class="p">[:</span><span class="mi">6</span><span class="p">,</span> <span class="p">:</span><span class="mi">6</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">smat</span><span class="p">,</span> <span class="n">vh</span><span class="p">)))</span>
<span class="go">True</span>
</pre></div>
</div>
<p>Reconstruction based on reduced SVD, 2D case:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vh</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">full_matrices</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">vh</span><span class="o">.</span><span class="n">shape</span>
<span class="go">((9, 6), (6,), (6, 6))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">u</span> <span class="o">*</span> <span class="n">s</span><span class="p">,</span> <span class="n">vh</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">smat</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">smat</span><span class="p">,</span> <span class="n">vh</span><span class="p">)))</span>
<span class="go">True</span>
</pre></div>
</div>
<p>Reconstruction based on full SVD, 4D case:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vh</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">full_matrices</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">vh</span><span class="o">.</span><span class="n">shape</span>
<span class="go">((2, 7, 8, 8), (2, 7, 3), (2, 7, 3, 3))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">*</span> <span class="n">s</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="p">:],</span> <span class="n">vh</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">],</span> <span class="n">s</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">vh</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
<p>Reconstruction based on reduced SVD, 4D case:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">vh</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">full_matrices</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">u</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">vh</span><span class="o">.</span><span class="n">shape</span>
<span class="go">((2, 7, 8, 3), (2, 7, 3), (2, 7, 3, 3))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">u</span> <span class="o">*</span> <span class="n">s</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="p">:],</span> <span class="n">vh</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="n">vh</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>

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